ew031 Green Lantern of Gotham reads:

Let's say he starts the turn with two tokens, so he can use barrier as a free action, so let's say he does. Let's say that then a)he loses a token by leadership, or b)he clears action tokens by the end of the turn. Does he loses barrier? Are the counters gone??

I'd say yes, but the trait says he gets it until your next turn, not "as long as he has two tokens", so thay's why I'm stumped

If you use that free action Barrier, those Barrier-markers will stay until the beginning of your next turn. It does not matter at all what happens to the Green Lantern's action tokens after the power has been used.